The magnetic Laplacian on the Disc for strong magnetic fields

TL;DR

This paper analyzes the eigenvalues of the magnetic Laplacian on a disc under strong magnetic fields, deriving asymptotics with precise estimates, and extends results to different boundary conditions using a variational approach.

Contribution

It provides new asymptotic formulas for eigenvalues of the magnetic Laplacian on a disc with strong magnetic fields, applicable to both Neumann and Dirichlet boundary conditions.

Findings

Eigenvalues asymptotics with exponentially small remainders

Extension of results to Dirichlet boundary condition

Variational method applicable to spectral analysis

Abstract

The magnetic Laplacian on a planar domain under a strong constant magnetic field has eigenvalues close to the Landau levels. We study the case when the domain is a disc and the spectrum consists of branches of eigenvalues of one dimensional operators. Under Neumann boundary condition and strong magnetic field, we derive asymptotics of the eigenvalues with accurate estimates of exponentially small remainders. Our approach is purely variational and applies to the Dirichlet boundary condition as well, which allows us to recover recent results by Baur and Weidl.